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Numerical Analysis of Partial Differential Equations

Physical phenomena are usually modeled by partial different equations. Hence, numerical analysis of partial different equations can be identified as one of the core areas in applied mathematics. Numerical solvers have to be carefully adapted to the specific functional analytic properties of the associated differential operator, leading to a whole zoo of different approaches.

Besides classical finite element methods, recently wavelet orthonormal bases have been used to derive provably optimal solvers for elliptic PDEs. This success story led to a very promising novel class of approaches to develop efficient solvers based on discretizations by systems from applied harmonic analysis. The first stage were wavelet frames, allowing the expoitation of their flexibility. Recently, ridgelets were successfully used in the discretization of linear transport equations. At the moment, partial differential equations whose solution exhibit curvilinear singularities are studied as the next step. For this, shearlet-type systems are anticipated to lead to optimal solvers.

Some of our Research Topics

We use systems from applied harmonic analysis, in particular, shearlets, for developing and analyzing efficient solvers for the following types of partial differential equations: